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Proof step 4 from proof heap-order-add-parent-key (previous step)

kar[m1 / 2] = key-index(key, m), n = m1 + 1, n < # kar, kar[n, key-index(key, m)][m0].key < kar[n, key-index(key, m)][(m0 - 1) / 2].key, m0 < # kar, m0 ≤ n, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ m0 = 0, n = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
kar[m1 / 2] = key-index(key, m), m0 < # kar, kar[m1 + 1, key-index(key, m)][m0].key < kar[m1 + 1, key-index(key, m)][(m0 - 1) / 2].key, m1 + 1 < # kar, m0 ≤ m1 + 1, ∀ m2. ¬ m1 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦ m0 = 0

Rule argument

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)